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Chapter 8: The Geometry of Vector Spaces

Expert-verified
Linear Algebra and its Applications
Pages: 437 - 483
Linear Algebra and its Applications

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

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138 Questions for Chapter 8: The Geometry of Vector Spaces

  1. Question 10: Find an example of a closed convex set \(S\) in \({\mathbb{R}^2}\) such that its profile P is nonempty but \({\mathop{\rm conv}\nolimits} P \ne S\).

    Found on Page 437

  2. In Exercises 9 and 10, mark each statement True or False. Justify each answer.

    Found on Page 437

  3. Question: In Exercise 10, let Hbe the hyperplane through the listed points. (a) Find a vector n that is normal to the hyperplane. (b) Find a linear functional f and a real number d such that \(H = \left( {f:d} \right)\).

    Found on Page 437

  4. The “B” in B-spline refers to the fact that a segment \({\bf{x}}\left( t \right)\)may be written in terms of a basis matrix, \(\,{M_S}\) , in a form similar to a Bézier curve. That is,

    Found on Page 437

  5. Repeat Exercise 9 for the points \({{\bf{q}}_{\bf{1}}}\),….\({{\bf{q}}_{\bf{5}}}\) whose barycentric coordinates with respect to S are given by \(\left( {\frac{{\bf{1}}}{{\bf{8}}},\frac{{\bf{1}}}{{\bf{4}}},\frac{{\bf{1}}}{{\bf{8}}},\frac{{\bf{1}}}{{\bf{2}}}} \right)\), \(\left( {\frac{{\bf{3}}}{{\bf{4}}}, - \frac{{\bf{1}}}{{\bf{4}}},{\bf{0}},\frac{{\bf{1}}}{{\bf{2}}}} \right)\),\(\left( {{\bf{0}},\frac{{\bf{3}}}{{\bf{4}}},\frac{{\bf{1}}}{{\bf{4}}},{\bf{0}}} \right)\),\(\left( {{\bf{0}}, - {\bf{2}},{\bf{0}},{\bf{3}}} \right)\), and \(\left( {\frac{{\bf{1}}}{{\bf{3}}},\frac{{\bf{1}}}{{\bf{3}}},\frac{{\bf{1}}}{{\bf{3}}},{\bf{0}}} \right)\), respectively.

    Found on Page 437

  6. Question: Find an example of a closed convex set S in \({\mathbb{R}^{\bf{2}}}\) such that its profile P is nonempty but \({\bf{conv}}\,P \ne S\).

    Found on Page 437

  7. Question: Let \({\bf{p}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{3}}}\\{\bf{1}}\\{\bf{2}}\end{array}} \right)\), \({\bf{n}} = \left( {\begin{array}{*{20}{c}}{\bf{2}}\\{\bf{1}}\\{\bf{5}}\\{ - {\bf{1}}}\end{array}} \right)\), \({{\bf{v}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{1}}\\{\bf{1}}\\{\bf{1}}\end{array}} \right)\), \({{\bf{v}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{ - {\bf{2}}}\\{\bf{0}}\\{\bf{1}}\\{\bf{3}}\end{array}} \right)\), and \({{\bf{v}}_{\bf{3}}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{4}}\\{\bf{0}}\\{\bf{4}}\end{array}} \right)\), and let H be the hyperplane in\({\mathbb{R}^{\bf{4}}}\) with normal n and passing through p. Which of the points \({{\bf{v}}_{\bf{1}}}\), \({{\bf{v}}_{\bf{2}}}\), and \({{\bf{v}}_{\bf{3}}}\) are on the same side of H as the origin, and which are not?

    Found on Page 437

  8. Explain why any set of five or more points in \({\mathbb{R}^3}\) must be affinely dependent.

    Found on Page 437

  9. Question: In Exercises 11 and 12, mark each statement True or False. Justify each answer.

    Found on Page 437

  10. Show that a set\(\left\{ {{{\bf{v}}_{\bf{1}}},...,{{\bf{v}}_p}} \right\}\)in\({\mathbb{R}^{\bf{n}}}\)is affinely dependent when \(p \ge n + 2\).

    Found on Page 437

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